Square Root of 94

The square root is the inverse of the square of the number. 94 is not a perfect square. The square root of 94 is expressed in both radical and exponential form.

In the radical form, it is expressed as √94, whereas (94)^(1/2) in the exponential form. √94 = 9.69536, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 94 is broken down into its prime factors.

Step 1: Finding the prime factors of 94 Breaking it down, we get 2 x 47, which are 2^1 x 47^1

Step 2: Now we found out the prime factors of 94. The second step is to make pairs of those prime factors. Since 94 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 94 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 94, we need to consider it as 94 itself.

Step 2: Now we need to find n whose square is 81. We can say n as ‘9’ because 9 x 9 is lesser than or equal to 94. Now the quotient is 9, and after subtracting 81 from 94, the remainder is 13.

Step 3: Now let us bring down 00 to make it 1300 as the new dividend. Add the old divisor with the same number 9 + 9 to get 18, which will be our new divisor.

Step 4: The new divisor will be 18n, where we need to find the value of n such that 18n x n ≤ 1300. Let us consider n as 7, now 187 x 7 = 1309, which is too large, so we try n as 6.

Step 5: Subtract 187 x 6 = 1122 from 1300, the difference is 178, and the quotient is 9.6.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 17800.

Step 7: Find the new divisor which is 192 (from 186 and 6), because 1926 x 6 = 11556. Step 8: Subtract 11556 from 17800, we get the result 6244.

Step 9: Now the quotient is 9.69. Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √94 is approximately 9.69.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 94 using the approximation method.

Step 1: Now we have to find the closest perfect square of √94. The smallest perfect square less than 94 is 81, and the largest perfect square greater than 94 is 100. √94 falls somewhere between 9 and 10.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (94 - 81) ÷ (100 - 81) = 13/19 ≈ 0.684 Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number which is 9 + 0.684 = 9.684, so the square root of 94 is approximately 9.684.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal, for example: 7.86, 8.65, and 9.42 are decimals.

• Long division method: A method used to find the square root of a non-perfect square by dividing and averaging, leading to an approximate value with decimals.